The standard uncertainty in the measurement theory is applied to evaluate the change of the rolling bearing vibration acceleration generated by the failure on the surface of the ring raceway. The time series are obtained via the experimental investigation on the vibrational acceleration of the rolling bearings with different failure diameters. And the result shows that the standard uncertainty of the vibrational acceleration increases nonlinearly with the failure diameter, revealing a new characteristic of the variation of the rolling bearing failure process. It follows that for a rolling bearing in running, the failure process can be described by the standard uncertainty of its vibration acceleration, laying a foundation for failure warning of a rolling bearing.